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DSP-CIS Chapter-8: Maximally Decimated PR-FBs

DSP-CIS Chapter-8: Maximally Decimated PR-FBs. Marc Moonen Dept. E.E./ESAT, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be / scd /. Part-III : Filter Banks. : Preliminaries Filter bank (FB) set-up and applications Perfect reconstructio n (PR) problem

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DSP-CIS Chapter-8: Maximally Decimated PR-FBs

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  1. DSP-CISChapter-8: Maximally Decimated PR-FBs Marc Moonen Dept. E.E./ESAT, KU Leuven marc.moonen@esat.kuleuven.be www.esat.kuleuven.be/scd/

  2. Part-III : Filter Banks : Preliminaries • Filter bank (FB) set-up and applications • Perfect reconstruction (PR) problem • Multi-rate systems review (=10 slides) • Example: Ideal filter bank (=10 figures) : Maximally Decimated PR-FBs • Example: DFT/IDFT filter bank • Perfect reconstruction theory • Paraunitary PR-FBs : DFT-Modulated FBs • Maximally decimated DFT-modulated FBs • Oversampled DFT-modulated FBs : Special Topics • Cosine-modulated FBs • Non-uniform FBs & Wavelets • Frequency domain filtering Chapter-7 Chapter-8 Chapter-9 Chapter-10

  3. synthesis bank analysis bank 3 3 F0(z) subband processing H0(z) OUT IN 3 3 F1(z) subband processing H1(z) + 3 3 F2(z) subband processing H2(z) downsampling/decimation upsampling/expansion 3 3 F3(z) subband processing H3(z) Refresh (1) General `subband processing’ set-up (Chapter-7) : PS: subband processing ignored in filter bank design

  4. 3 3 F0(z) subband processing H0(z) OUT IN 3 3 F1(z) subband processing H1(z) + 3 3 F2(z) subband processing H2(z) 3 3 F3(z) subband processing H3(z) Refresh (2) PS: analysis filters Hi(z) are also decimation/anti-aliasing filters, to avoid aliasing in subband signals after decimation (downsampling) PS: synthesis filtersGi(z) are also interpolation filters, to remove images after expanders (upsampling) downsampling/decimation upsampling/expansion

  5. 3 3 F0(z) subband processing H0(z) OUT IN 3 3 F1(z) subband processing H1(z) + 3 3 F2(z) subband processing H2(z) 3 3 F3(z) subband processing H3(z) Refresh (3) Assume subband processing does not modify subband signals (e.g. lossless coding/decoding) The overall aim would then be to have y[k]=u[k-d], i.e. that the output signal is equal to the input signal up to a certain delay But: downsampling introduces ALIASING… y[k]=u[k-d]?

  6. 3 3 F0(z) subband processing H0(z) OUT IN 3 3 F1(z) subband processing H1(z) + 3 3 F2(z) subband processing H2(z) 3 3 F3(z) subband processing H3(z) Refresh (4) Question : Can y[k]=u[k-d] be achieved in the presence of aliasing ? Answer : Yes !! PERFECT RECONSTRUCTION banks with synthesis bank designed to remove aliasing effects ! y[k]=u[k-d]?

  7. Refresh (5) Two design issues : ✪Filter specifications, e.g. stopband attenuation, passband ripple, transition band, etc. (for each (analysis) filter!) ✪Perfect reconstruction (PR) property. Challenge will be in addressing these two design issues at once (‘PR only’(without filter specs) is easy, see next slides) This chapter : Maximally decimated FB’s :

  8. Example: DFT/IDFT Filter Bank • Basic question is..: Downsampling introduces ALIASING, then how can PERFECT RECONSTRUCTION (PR) (i.e. y[k]=u[k-d]) be achieved ? • Next slides provide simple PR-FB example, to demonstrate that PR can indeed (easily) be obtained • Discover the magic of aliasing-compensation…. y[k]=u[k-d]? 4 4 G1(z) output = input H1(z) u[k] 4 4 G2(z) output = input H2(z) + 4 4 output = input G3(z) H3(z) 4 4 G4(z) output = input H4(z)

  9. u[k] u[0],0,0,0,u[4],0,0,0,... 4 4 4 u[-1],u[0],0,0,u[3],u[4],0,0,... 4 4 4 u[-2],u[-1],u[0],0,u[2],u[3],u[4],0,... + + + 4 4 u[-3],u[-2],u[-1],u[0],u[1],u[2],u[3],u[4],... u[k-3] Example: DFT/IDFT Filter Bank First attempt to design a perfect reconstruction filter bank - Starting point is this : convince yourself that y[k]=u[k-3] …

  10. + 4 4 u[k] 4 4 u[k-3] 4 4 4 4 Example: DFT/IDFT Filter Bank - An equivalent representation is ... As y[k]=u[k-d], this can already be viewed as a (1st)perfect reconstruction filter bank (with lots of aliasing in the subbands!) All analysis/synthesis filters are seen to be pure delays, hence are not frequency selective (i.e. far from ideal case with ideal bandpass filters….) PS: Transmux version (TDM) see Chapter-7

  11. 4 4 u[k] 4 4 u[k-3] + 4 4 4 4 Example: DFT/IDFT Filter Bank -now insert DFT-matrix (discrete Fourier transform) and its inverse (I-DFT)... as this clearly does not change the input-output relation (hence perfect reconstruction property preserved)

  12. 4 4 u[k-3] 4 4 + 4 4 4 4 Example: DFT/IDFT Filter Bank - …and reverse order of decimators/expanders and DFT-matrices (not done in an efficient implementation!) : =analysis filter bank =synthesis filter bank This is the `DFT/IDFT filter bank’. It is a first (or 2nd) example of a maximally decimated perfect reconstruction filter bank ! u[k]

  13. u[k] Example: DFT/IDFT Filter Bank What do analysis filters look like? This is seen (known) to represent a collection of filters Ho(z),H1(z),..., each of which is a frequency shifted version of Ho(z) : i.e. the Hi are obtained by uniformly shifting the `prototype’ Ho over the frequency axis.

  14. Example: DFT/IDFT Filter Bank H1(z) Ho(z) H2(z) H3(z) The prototype filter Ho(z) is a not-so-great lowpass filter with first sidelobe only 13 dB below the main lobe. Ho(z) and Hi(z)’s are thus far from ideal lowpass/ bandpass filters. Hence (maximal) decimation introduces significant ALIASING in the decimated subband signals Still, we know this is a PERFECT RECONSTRUCTION filter bank (see construction previous slides), which means the synthesis filters can apparently restore the aliasing distortion. This is remarkable! Other perfect reconstruction banks : read on.. N=4

  15. + Example: DFT/IDFT Filter Bank Synthesis filters ? synthesis filters are (roughly) equal to analysis filters… PS: Efficient DFT/IDFT implementation based on FFT algorithm (`Fast Fourier Transform’). *(1/N) N=4

  16. Perfect Reconstruction Theory Now comes the hard part…(?) • 2-channel case: Examples… ✪ M-channel case: Polyphase decomposition based approach

  17. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case It is proved that...(try it!) • U(-z) represents aliased signals, hence A(z) referred to as `alias transfer function’’. • T(z) referred to as `distortion function’ (amplitude & phase distortion).

  18. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case • Requirement for `alias-free’ filter bank : If A(z)=0, then Y(z)=T(z).U(z), hence the complete filter bank behaves as a linear time invariant (LTI) system (despite up- & downsampling) !!!! • Requirement for `perfect reconstruction’ filter bank (= alias-free + distortion-free): i) ii)

  19. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Perfect Reconstruction : 2-Channel Case • A first attempt is as follows….. : so that For the real coefficient case, i.e. which means the amplitude response of H1 is the mirror image of the amplitude response of Ho with respect to the quadrature frequency hence the name `quadrature mirror filter’ (QMF)  

  20. y[k] u[k] 2 2 F0(z) H0(z) + 2 2 F1(z) H1(z) Ho H1 Perfect Reconstruction : 2-Channel Case `quadrature mirror filter’ (QMF) : hence if Ho (=Fo) is designed to be a good lowpass filter, then H1 (=-F1) is a good high-pass filter.

  21. Perfect Reconstruction : 2-Channel Case • A 2nd (better) attempt is as follows: [Smith & Barnwell 1984] [Mintzer 1985] i) so that (alias cancellation) ii) `power symmetric’ Ho(z) (real coefficients case) iii) so that (distortion function) ignore the details! This is a so-called`paraunitary’ perfect reconstruction bank (see below), based on a lossless system Ho,H1 :   This is already pretty complicated…

  22. 4 4 F0(z) H0(z) u[k] y[k] 4 4 F1(z) H1(z) + 4 4 F2(z) H2(z) 4 4 F3(z) H3(z) Perfect Reconstruction : M-Channel Case It is proved that...(try it!) • 2nd term represents aliased signals, hence all `alias transfer functions’ Al(z) should ideally be zero(for all l ) • T(z) is referred to as `distortion function’ (amplitude & phase distortion). For perfect reconstruction, T(z) should be a pure delay Sigh !!…

  23. 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case • A simpler analysis results from a polyphase description : i-th row of E(z) has polyphase components of Hi(z) i-th column of R(z) has polyphase components of Fi(z) u[k] u[k-3] Do not continue until you understand how formulae correspond to block scheme!

  24. u[k] u[k-3] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case • With the `noble identities’, this is equivalent to: Necessary & sufficient conditions for i) alias cancellation ii) perfect reconstruction are then derived, based on the product

  25. u[k] u[k-3] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case Necessary & sufficient condition for alias-free FB is…: a pseudo-circulant matrix is a circulant matrix with the additional feature that elements below the main diagonal are multiplied by 1/z, i.e. ..and first row of R(z).E(z) are polyphase cmpnts of `distortion function’ T(z) read on->

  26. u[k] 4 4 4 4 + + 4 4 4 4 Perfect Reconstruction : M-Channel Case PS: This can be explained as follows: first, previous block scheme is equivalent to (cfr. Noble identities) then (iff R.E is pseudo-circ.)… so that finally.. 4 4 4 4 T(z)*u[k-3] 4 4 u[k] 4 4

  27. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Perfect Reconstruction : M-Channel Case Necessary & sufficient condition for PR is…: (i.e. where T(z)=pure delay, hence p_r(z)=pure delay, and all other p_i(z)=0) I_n is nxn identity matrix, r is arbitrary Example (r=0) : for conciseness, will use this from now on !

  28. Perfect Reconstruction : M-Channel Case • Example : DFT/IDFT Filter bank : E(z)=F , R(z)=F^-1 • Design Procedure: 1. Design all analysis filters (see Part-II). 2. This determines E(z) (=polyphase matrix). 3. Assuming E(z) can be inverted (?), synthesis filters are (delta to make synthesis causal) • Will consider only FIR analysis filters, leading to simple polyphase decompositions (see Chapter-7). However, FIR E(z) generally leads to IIR R(z), where stability is a concern…

  29. Perfect Reconstruction : M-Channel Case PS: Inversion of matrix transfer functions ?… • The inverse of a scalar (i.e. 1-by-1 matrix) FIR transfer function is always IIR (except for contrived examples) • The inverse of an N-by-N (N>1) FIR transfer function can be FIR

  30. Perfect Reconstruction : M-Channel Case PS: Inversion of matrix transfer functions ?… Compare this to inversion of integers and integer matrices: • The inverse of an integer is always non-integer (except for `E=1’) • The inverse of an N-by-N (N>1) integer matrix can be integer

  31. Perfect Reconstruction : M-Channel Case Question: Can we build FIR matrices E(z) that have an FIR inverse? Answer: YES, `Unimodular’ matrices = matrices with {determinant=constant*z^-d} e.g. where the Ei’s are constant (≠ a function of z) invertible matrices Example:E(z) = FIR LPC lattice , see Chapter-5 (not a good filter bank though… explain!) Design Procedure: optimize Ei’s to obtain filter specs (ripple, etc.), etc..

  32. Perfect Reconstruction : M-Channel Case Question: Can we avoid direct inversion, e.g. build FIR E(z) matrices with additional `special properties’, so that R(z) is trivially obtained and its specs are better controlled? (compare with (real) orthogonal or (complex) unitary matrices, where inverse is equal to (Hermitian) transpose) Answer: YES, `paraunitary’matrices (=special class of FIR matrices with FIR inverse) See next slides…. Will focus on paraunitary E(z) leading to PR/FIR/paraunitaryfilter banks

  33. Paraunitary PR Filter Banks Review : `PARACONJUGATION’ • For a scalar transfer function H(z), paraconjugate is i.e it is obtained from H(z) by - replacing z by 1/z - replacing each coefficient by its complex conjugate Example : On the unit circle, paraconjugate corresponds to complex conjugate paraconjugate = `analytic extension’ of unit-circle complex conjugate

  34. Paraunitary PR Filter Banks Review : `PARACONJUGATION’ • For a matrix transfer function H(z), paraconjugate is i.e it is obtained from H(z) by - transposition - replacing z by 1/z - replacing each coefficient by is complex conjugate Example : On the unit circle, paraconjugate corresponds to conjugate transpose(*) paraconjugate = `analytic extension’ of unit-circle conjugate transpose(*) (*) =Hermitian transpose

  35. Paraunitary PR Filter Banks Review : `PARAUNITARY matrix transfer functions’ • Matrix transfer function H(z), is paraunitary if (possibly up to a scalar) For a square matrix function A paraunitary matrix is unitary on the unit circle paraunitary = `analytic extension’ of unit-circle unitary. PS: if H1(z) and H2(z) are paraunitary, then H1(z).H2(z) is paraunitary

  36. u[k-3] u[k] 4 4 4 4 + 4 4 4 4 Paraunitary PR Filter Banks - If E(z) is paraunitary then perfect reconstruction is obtained with (delta to make synthesis causal) If E(z) is FIR, then R(z) is also FIR !! (cfr. definition paraconjugation)

  37. Paraunitary PR Filter Banks • Question: Can we build paraunitary FIR E(z) (with FIR inverse R(z))? • Answer: Yes! where the Ei’s are constant unitary matrices • Example: 1-input/2-output FIR lossless lattice, see Chapter-5 Example: 1-input/M-output FIR lossless lattice, see Chapter-5 • Design Procedure: optimize unitary Ei’s (e.g. rotation angles in lossless lattices) to obtain analysis filter specs, etc..

  38. Paraunitary PR Filter Banks Properties of paraunitary PR filter banks: (proofs omitted) • If polyphase matrix E(z) (and hence E(z^N)) is paranunitary, and then vector transfer function H(z) (=all analysis filters ) is paraunitary • If vector transfer function H(z) is paraunitary, then its components are power complementary (lossless 1-input/N-output system) (see Chapter 5)

  39. Paraunitary PR Filter Banks Properties of paraunitary PR filter banks (continued): • Synthesis filters are obtained from analysis filters by conjugating the analysis filter coefficients + reversing the order (cfr page 34): • Hence magnitude response of synthesis filter Fk is the same as magnitude response of corresponding analysis filter Hk: • Hence, as analysis filters are power complementary (cfr. supra), synthesis filters are also power complementary • Examples: DFT/IDFT bank, 2-channel case (p. 21) • Great properties/designs....

  40. Conclusions • Have derived general conditions for perfect reconstruction, based on polyphase matrices for analysis/synthesis bank • Seen example of general PR filter bank design : PR/FIR/Paraunitary FBs, e.g. based on FIR lossless lattice filters • Sequel = other (better) PR structures Chapter 9: Modulated filter banks Chapter 10: Oversampled filter banks, etc.. • Reference: `Multirate Systems & Filter Banks’ , P.P. Vaidyanathan Prentice Hall 1993.

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